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Get the roots of a derivative for applying the Lucas–Gauss theorem on a set of points

Contributed by:
Ed Pegg Jr

ResourceFunction["PointsetDerivativeRoots"][ applies the Lucas-Gauss derivative on a set of |

The Lucas–Gauss theorem states that the convex hull of the roots of any nonconstant complex polynomial contains the roots of its derivative.

With input {{*r*_{1},*im*_{1}},{*r*_{2},*im*_{2}},…} and *z*_{i}=*r*_{i}+ⅈ*im*_{i}, the roots of derivative D[(z-z_{1})(z-z_{2})…,z] are returned as points.

Throughout this document, the roots of a derivative of a nonconstant complex polynomial generated by points are called "roots", out of convenience.

Find the roots of a set of points:

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By Marden's theorem, the roots of triangle vertices are the foci of an ellipse tangent to midpoints of the triangle's edges:

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Find the roots of a set of points:

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By the Lucas–Gauss theorem, the convex hull of the roots is entirely contained by the convex hull of the original points:

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The root of two points is the midpoint:

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The root of three points is given by the foci of the ellipse tangent to the midpoints of the triangle:

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This is equivalent to the Steiner circumellipse, scaled by a factor of 1/2:

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Much as in a convex hull, duplicating a point will not affect the result:

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Show eleven levels of the Lucas-Gauss theorem:

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- MathWorld - Lucas's Root Theorem
- Wikipedia - Gauss–Lucas theorem
- Wolfram Demonstrations - Lucas-Gauss Theorem

- 1.0.0 – 08 March 2023

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